In this section, we specialize the constructions of Section 5.2 to the case of generic rational surfaces. In agreement with the discussion ending that section, we show that the genericity condition for surfaces is precisely the combinatorial normal crossing boundary condition plus a tangency condition at crossing points. As stated in Proposition 5.2.10, we show that the ambient space need not be smooth.
We start by describing our input, a polynomial parameterization f : T2 → Tn where
n ≥ 3, f = (f1, . . . , fn) and fi ∈ C[x±11 , x ±1
2 ] for all i = 1, . . . , n. We let Y be the Zariski
closure of the image of f in Tn. We assume our map f is generically finite, so Y is a surface
in Tn. Our goal is to compute the tropical surface graph associated to Y .
Following Khovanski˘ı’s philosophy [35, 57], we assume each fi is generic relative to its
support. We now state the main result in this section. The remainder of the section will be devoted to its proof and to give several numerical examples. For simplicity, we assume that our choices of coefficients give irreducible polynomials. The genericity condition says that the plane curves defined by the polynomials fi satisfy the combinatorial normal crossing
Theorem 5.3.1. Let f = (f1, . . . , fn) : T2 → Tn be a generically finite map. Assume that
the polynomials fi are irreducible and generic relative to their Newton polytopes Pi, i.e. no
three plane curves in T2 defined by these n polynomials intersect at a point and the tangent
directions at pairwise crossing points of the branches of two curves are distinct. Then, the tropical surface graph associated to the variety im f ⊂ Tn can be described as follows. Let
N be the common refinement of the n inner normal fans of the polytopes Pi (i = 1, . . . , n),
and let ρ1, . . . , ρl be the rays of N , oriented counterclockwise, with primitive generators nρj
for all j = 1, . . . , l. With this notation, the nodes of the graph are
{ei : dim Pi 6= 0, 1 ≤ i ≤ n} ∪ {[Dρ] := trop(f )(nρ) : [Dρ] 6= 0, ρ ∈N [1]}.
The list of edges and their weights is:
(i) m([Dρj],[Dρk]) = δ−1| gcd 2 × 2 − minors ([Dρj] | [Dρk])|/| det(nρk | nρj)|, if |j − k| = 1
mod l or 0 if not. (ii) m(ei,[Dρ]) = δ −1(|face nρ(Pi) ∩ Z 2| − 1) gcd [D ρ]j : j 6= i, if nρ∈ T (fi), or 0 if not. (iii) m(ei,ej) = δ −1length((f
i = fj = 0) ∩ T2) if dim(Pi + Pj) = 2, and 0 if not. If the
coefficients are generic enough, this number equals 1/δ times the mixed volume of the polytopes Pi and Pj.
It is important to point out that the previous algorithm was already presented in [92] and further studied in [91]. We contribute to the subject by elucidating the right genericity condition to impose. The proof of [92, Theorem 2.1] requires the genericity of both the coefficients and the Newton polytopes, to ensure that the Minkowski sum of the n polytopes P1, . . . , Pn is a simple smooth polytope. Our proof discards this extra assumption on the
polytopes, unraveling the key aspects in their argumentation. In addition, we correct the missing factor of 1/δ in [92], following [91, Theorem 5.1].
Proof. As we stated in the previous section, our task is to compactify Y ⊂ Tn. Instead, we work with X := f−1(Y ) = T2
r Sni=1(fi = 0). Using the knowledge of the Newton
polytopes P1, . . . , Pn, we construct a projective toric variety P(Σ) and compactify X inside
this space [92, Theorem 2.1]. We now explain how to construct the fan Σ from these n input polynomials. Consider the Minkowski sum of the n polytopes P1, . . . , Pn. Since the
polynomials are generic, its inner normal fan is the common refinement of the n normal fans of these polytopes. We letN be this fan. Notice that the fan need not be strictly simplicial, thus P(N ) need not be a smooth toric surface. To fix this, we perform a refinement of the fan N by a strictly simplicial fan N 0. This is done by subdividing the two-dimensional cones
that are not unimodular [15, Section 5]. On the geometric side, this operation corresponds to performing toric blow-ups on the surface P(N ): we blow-up torus invariant points associated to these two-dimensional cones. The output is a smooth projective toric surface P(N 0) that compactifies X. We set Σ to be the fan N 0.
The boundary of P(N 0) consists of two types of divisors. The first class of divisors are the toric divisors Dρ indexed by the rays ρ in N 0. They correspond to facets of the Minkowski
sum Pn
i=1Pi. The toric boundary
S
ρ∈N0[1]Dρ has simple normal crossings because N 0 is
a strictly simplicial fan. Similarly, the toric divisor S
ρ∈N[1]Dρ has combinatorial normal
crossings becauseN is a simplicial fan.
The second type of components consist of n divisors E1, . . . , En, which are the closures
in P(N ) of the divisors Ei = (fi = 0) ⊂ T2 in X. The irreducibility of the polynomials
fi and the proof of [92, Theorem 2.1] show that these divisors are smooth and irreducible
and that the union of all Dρ’s and all Ej’s has simple normal crossings. Notice that if fj
consists of a single monomial, then Ej is the empty set. Such indices will not induce a node
in the boundary intersection complex of X, so from now on we may assume dim Pi > 0 for
all i = 1, . . . , n.
Thus, the boundary divisor of X and its smooth compactification P(N 0) can be decom- posed as
∂ := P(N 0) r X = D1∪ . . . ∪ Dl∪ Dl+1∪ . . . Dm∪ E1∪ . . . ∪ En,
where ρ1, . . . , ρl are the rays of N , ρl, . . . , ρm are the rays in N 0[1]r N [1] and Di denotes
the toric divisor Dρi (i = 1, . . . , m).
The simplicial complex ∆P(N0),∂ is a graph with m + n vertices. The edges of this graph
consist of pairs of vertices C ∪ J where C ⊂ {1, . . . , m} and J ⊂ {1, . . . , n} and |C| + |J | = 2. Following [92], we denote by ∆P(N0),∂[J ] the subset of edges in ∆
P(N0),∂ with fixed J . By
construction, we have three possibilities: |J | = 0, 1 or 2. For |J | = 0, the edges ∆P(N0),∂[J ]
are of the form (Dρ, Dρ0) for ρ and ρ0 rays in the fan N 0. By standard intersection theory
on smooth toric varieties, we know that the intersection numbers among the torus-invariant divisors Dρ are:
Dρ· Dρ0 =
(
1 if ρ and ρ0 generate a two-dimensional cone in N 0,
0 else. (5.7)
In particular, this says that we only have edges among consecutive rays of N 0, if counter- clockwise oriented.
When |J | = 1, we seek to identify edges of the form (Ej, Dρ), for ρ ∈N 0and j = 1, . . . , n.
Again, this is done by toric methods. Since Ej represents a Cartier divisor with local
equation fj and Dρ is a torus invariant divisor, the intersection number will correspond to
the intersection number of the initial form innρ(fj) and Dρ. This quantity coincides with the
number of nonzero solutions of a univariate polynomial. The Newton polytope of innρ(fj)
agrees with facenρ(Pj). Since nρ6= 0, this polytope has dimension zero or one. If this number
is zero, the initial form is a monomial, and the intersection number is zero. If the dimension is one, Newton-Puiseux’s theorem implies that the intersection number is the lattice length of the edge facenρ(Pj). Thus, we see that Ej is adjacent to a node Dρ if and only if nρ is
a ray in the normal cone of the polytope Pj, or equivalently, if it belongs to the tropical
hypersurface T (fj). In addition:
Ej· Dρ= lattice length of facenρ(Pj) = |facenρ(Pj) ∩ Z
2| − 1. (5.8)
Finally, if |J | = 2, we want to certify which edges (Ei, Ej) belong to the boundary
complex. We claim it suffices to check if the equations fi and fj have a common root in
T2. In fact, the remaining intersection points outside the big open torus will lie in the toric boundary of P(N 0) and, thus, will yield a triple intersection among the divisors Ei, Ej
and some toric divisor Dρ. Therefore, the intersection number is the length of the zero-
dimensional scheme (fi = fj = 0) ∩ T2. If the coefficients of these polynomials are generic,
Bernstein’s theorem show that this length equals the mixed volume of the polytopes Pi
and Pj, whereas for special choices of coefficients, this number is an upper bound for the
true intersection number [3, Theorem A]. The mixed volume is nonzero if and only if the Minkowski sum of the corresponding polytopes is two-dimensional. Therefore, the edges of the form (Ei, Ej) in the boundary complex must satisfy dim(Pi+ Pj) = 2.
Notice that since we are interested in the realization of the boundary complex, we can safely assume that the dimension restriction characterizes the edges (Ei, Ej). “Artificial”
edges in the abstract graph will have weight zero in its realization. For this reason, we safely add these artificial edges to the boundary complex and still obtain the correct tropical surface graph. Summarizing:
(Ei, Ej) is an edge of ∆P(N0),∂ ⇐⇒ dim(Pi+ Pj) = 2.
We now discuss the realization of the boundary complex. As we know, this is done by describing the divisorial valuation of each component of the boundary ∂. By construction, valE
j(fi) = δi,j, so the node corresponding to Ej maps to ej, the j
th element of the canonical
basis. We compute the divisorial valuation of all Dρ’s with the tools of toric geometry [39,
Section 5.2]. Without loss of generality, assume ρ = ρ1. We pick the primitive generator nρ
of ρ and we extend this vector to a Z-basis of R2. Thus, we can assume ρ = e
1. By definition,
the divisorial valuation valDρ is obtained from the order of vanishing of all the polynomials fj
at Dρ, that is, by the maximal exponent of the variable x1 dividing fj in the polynomial ring
C[x±12 ][x1]. Notice that this number can be negative. The maximum exponent is precisely the
minimum value among the inner products he1, νi, for all possible monomials with exponent
ν ∈ Z2 that appear in fj. Hence, f#([Dρ]) = valDρ(f
∗(χ
j)) = valDρ(fj) = trop(fj)(e1).
With the same reasoning, we obtain [Di] := (valDi(fj))
n
j=1= (trop(f1)(nρi), . . . , trop(fn)(nρi)) = trop(f )(nρi) ∀i = 1, . . . , m.
Using Corollary 5.2.5 and expressions (5.7) and (5.8), we compute the weights of the edges in the realization of ∆P(N0),∂:
(i) m([Dρ],[Dρ0]) = δ
−1gcd 2 × 2 − minors ([D
ρ] | [Dρ0])/| det gcd 2 × 2 − minors (nρ| nρ0)|,
if nρ, nρ0 span a two-dimensional cone in N 0. In other cases, this number is is 0.
(ii) m([E i],[Dρ]) = δ −1(|face nρ(Pi) ∩ Z 2| − 1) gcd trop(f j)(nρ) : j 6= i, if nρ ∈ T (fi), or 0 if not. (iii) m([E i],[Ej]) = δ −1length((f
i = fj = 0) ∩ T2) if dim(Pi+ Pj) = 2 or zero otherwise. If the
coefficients of fi, and fj are generic enough, this number is precisely the mixed volume
of the polytopes Pi and Pj, multiplied by a factor of 1/δ.
Notice that the statement of our theorem constructs the tropical surface graph by means of the fan N rather than our choice Σ = N 0. We now explain how can we avoid per- forming the refinement N 0 of the fan N in the previous argument, in agreement with the combinatorial normal crossing genericity condition required for the polynomials f1, . . . , fn
and the spirit of Proposition 5.2.10. First we provide set-theoretic reasons and then focus on the weights of the tropical surface graph. By induction on the number of elements in N 0[1]
r N [1], it suffices to show that subdividing a two-dimensional cone on N by adding a single ray gives the same tropical surface graph.
Our first goal is to show that the rays added toN give nodes and edges in the realization of the complex that are contained in cones over edges of the form ([Dρ], [Dρ0]) for consecutive
edges ρ, ρ0inN [1]with counterclockwise orientation. Each ray that we add toN corresponds
to a blow-up at a torus-invariant point p of P(N ). This point is the torus orbit associated to the two-dimensional cone spanned by ρ and ρ0 and supports the intersection of the two torus-invariant divisors, thought of as Q-Cartier divisors:
Dρ∩ Dρ0 = (gcd(2 × 2 − minors (nρ| nρ0)))−1p.
This implies Dρ· Dρ0 = (gcd(2 × 2 − minors (nρ| nρ0)))−1, where the intersection product is
considered in the (possibly singular) toric variety P(N ) [39, Section 5.1].
At each step of the toric resolution, we subdivide the cone spanned by consecutive rays ρ and ρ0ofN by adding a ray τ in this cone. Call N 0 the corresponding refinement. Note that this extra ray modifies the boundary intersection complex associated to P(N ) as follows. It adds a node Dτ and replaces the edge (Dρ, Dρ0) in the boundary complex associated to
P(N ) with two edges: (Dρ, Dτ) and (Dτ, Dρ0). The key observation is that the combinatorial
normal crossing condition on the boundary of P(N ) is preserve under refinements of the fan N . Notice that every ray that we add to N optimizes a single vertex of each Newton polytope, so it does not intersect any of the divisors Ei.
By construction, the valuations of the torus-invariant divisors associated to rays in N [1] viewed as divisors in P(N ) or P(N 0) are the same. If we write nτ = a nρ + b nρ0, for
a, b ∈ Q>0, we see that the divisorial valuation of Dτ equals a valDρ + b valDρ0. Thus, the
the edges ([Dρ], [Dτ]) and ([Dτ], [Dρ]), with no partial overlap between the last two cones.
If we write a = c/q and b = d/q with c, d, q ∈ Z, we see that gcd(2 × 2 − minors([Dτ] | [Dρ])) = b gcd(2 × 2 − minors([Dρ] | [Dρ0])), gcd(2 × 2 − minors([Dτ] | [Dρ0])) = a gcd(2 × 2 − minors([Dρ] | [Dρ0])), | det([nτ] | [nρ])| = b | det([nρ] | [nρ0])|, | det([nτ] | [nρ0])| = a | det([nρ] | [nρ0])|. (5.9)
When analyzing the intersection numbers among pairs of boundary divisors we only need to look at the three divisors Dρ, Dρ0 and Dτ. As we mentioned, the intersection of the two
divisors Dρand Dρ0 in P(N ) is replaced by the chain of intersections Dρ, Dτ, and Dτ, Dρ0 in
P(N 0). Using (5.9) these two intersection numbers in P(N 0) are related to the intersection number Dρ· Dρ0 in P(N ) by the expressions
Dρ· Dτ =
1
bDρ· Dρ0, Dτ · Dρ0 = 1
aDρ· Dρ0.
Combining these identities with Theorem 5.2.10, we see that the multiplicities of the edges ([Dρ], [Dτ]) and ([Dτ], [Dρ0]) in the realization of the boundary complex of P(N 0) equal the
multiplicity of the edge ([Dρ], [Dρ0]) in the realization of the boundary complex of P(N ).
This conclude our proof.
Remark 5.3.2. The last part of the previous proof shows the effect of the blow-ups on the realization of the boundary intersection complex for a given compactification: both weighted graphs in Rn yield the same weighted set T Y . This effect was discussed in the
proof of Proposition 5.2.10 and in the construction of the tropical secant surface graph of Section 4.3. It also remarks the key role of the combinatorial normal crossing boundary as the generic condition to ensure the validity of the conclusion in Theorem 5.3.1.
We illustrate the previous methods for computing tropicalizations of generic surfaces with three examples. In the next section, we revisit the last two examples for special choices of coefficients that violate the genericity condition. In particular, we show how the resulting tropical surface graphs need to be modified to reflect the non-genericity of these surfaces. Example 5.3.3. Our first example is a modification of [92, Example 3.4], where we remove a monomial factor from each polynomial to make them irreducible over the polynomial ring C[x, y, z]. This change will have no effect on the combinatorics of the tropical surface graph, but will change its coordinates. Consider the general surface in Y ⊂ T3 parameterized by
three bivariate polynomials f1(s, t) = a1+ a2s2t + a3st2, f2(s, t) = b1st + b2s + b3t, f3(s, t) = c1t + c2s2+ c3st2,
where a1, a2, a3, b1, b2, b3, c1, c2, c3 ∈ C are generic nonzero coefficients. By construction, the
map f has degree δ = 1. Their Newton polytopes and corresponding inner normal fans are depicted in Figures 5.1 and 5.2. All rays in these fans have multiplicity one, since the lattice lengths of the corresponding edges in the polytopes equal one.
P1 v1 v2 v3 (0, 0) (1, 2) (2, 1) P2 v1 v2 v3 (0, 1) (1, 0) (1, 1) P3 v1 v2 v3 (0, 1) (2, 0) (1, 2)
Figure 5.1: From left to right: Newton polytopes of the polynomials f1, f2 and f3.
N (P1) v1 v2 v3 (−1, −1) (2, −1) (−1, 2) N (P2) v1 v2 v3 (0, −1) (−1, 0) (1, 1) N (P3) v1 v2 v3 (−2, −1) (1, −1) (1, 2)
Figure 5.2: From left to right: inner normal fans of the Newton polytopes P1, P2 and P3.
Next, we compute the Minkowski sum P := P1+ P2+ P3 of the three Newton polytopes,
and its associated inner normal fan. This fan corresponds to the common refinement of the normal fans of the three constituent polytopes. Notice that the resulting fan N (P) is not strictly simplicial, since the maximal cones corresponding to the vertices v1, v7 and v9 of P
are not unimodular. Thus, we subdivide these three cones by adding the rays r19, r89 and
r67. The resulting fan N 0 has f -vector (1, 12, 12). The polytope P and the fan N 0 are
depicted in Figure 5.3.
Following Algorithm 5.1, we construct the nodes of the graph encoding T Y . To simplify notation, we denote by Di the toric divisor associated to the ray ri. The corresponding
primitive vectors nri are indicated in Figure 5.3. The nodes associated to the nine toric
divisors are: [D1] := (−2, −1, −2), [D4] := (−2, −1, −2), [D7] := (0, 1, 1), [D2] := (−5, −3, −4), [D5] := (−1, −1, −1), [D8] := (0, 1, 2), [D3] := (−3, −2, −3), [D6] := (0, −1, −1), [D9] := (0, −1, −2). (5.10)
P := L3 i=1 Pi
v
1v
2v
3v
4v
5v
6v
7v
8v
9 (2, 1) (2, 2) (1, 4) (0, 5) (−1, 5) (−2, 4) (−3, 2) (−2, 1) (0, 0) (−2, −1) (1, −1) (1, 2) (0, −1) (−1, 0) (1, 1) (−1, −1) (2, −1) (−1, 2) (0, 1) (−1, 1) (1, 0)v
1v
2v
3v
4v
5v
6v
7v
8v
9 N 0≺ W3 i=1N (Pi ) r2 r5 r8 r4 r1 r7 r3 r6 r9 r89 r19 r67Figure 5.3: From left to right: Minkowski sum P of P1, P2 and P3 and a strictly simplicial fanN 0
refining the normal fan of P. The different colors indicate the corresponding normal fans of each Pi
(i = 1, 2, 3). The dashed rays r19, r89 and r67are introduced to refine the singular cones ofN (P).
Ignoring these three rays gives the nine chambers in the normal fan of P, which are dual to the nine vertices of P.
Notice that [D1] = [D4], so the tropical surface graph in R3 has fewer nodes than expected:
there are eleven nodes in this realization. Likewise, some edges ([Dρ], ei) or ([Dρ], [Dη]) may
give one-dimensional cones in the tropical variety T Y . For this reason, in Algorithm 5.1, we tested the dimension of the corresponding cones before adding any of these pairs to the list of edges of our tropical surface graph. After computing these numbers, we obtain 19 edges: the three edges (ei, ej), the nine edges ([D3], e1), ([D6], e1), ([D9], e1), ([D1], e2), ([D4], e2),
([D7], e2), ([D2], e3), ([D5], e3) and ([D8], e3), the seven edges ([Di], [Di+1]) (i = 1, . . . , 8,
i 6= 6) and ([D9], [D1]).
The weights of these edges are computed using mixed volumes, and are indicated in the left-most picture in Figure 5.4. We start with the edges corresponding to |J | = 2 and C = {(0, 0)}, that is, to the three edges (ei, ej). These mixed volumes are obtained using the
formula MV(Pi, Pj) = Vol(Pi+ Pj) − Vol(Pi) − Vol(Pj), where Vol( ) denotes the Euclidean
volume in R2. Figure 5.4 shows the pairwise Minkowski sums of the polytopes P
1, P2, P3.
This gives m12 = 12(12 − 3 − 1) = 4, m23= 12(10 − 1 − 3) = 3, m12= 12(18 − 3 − 3) = 6.
We now consider the case where |J | = 1 and C is a ray in the fan N . These are edges of the form ([Dρ], ei), where ρ is a ray in the normal fan N (Pi) of the polytope Pi. Since
the edges of all these polytopes have lattice length one and the generating rays of N are primitive, our task reduces to computing the gcd of the maximal minors of the 3 × 2-matrices ([Dρ] | ei), that is, the gcd of all coordinates [Dρ] except for the ith one. All such numbers
equal one, except for the edges with ρ = r1 = r4 and i = 2, whose value is two. Since the
of the edge ([D1], e2), which equals four. This explains the transition from the left to the
right of Figure 5.5.
Finally we compute the values of the edges with |J | = 0, that is, edges of the form ([Dρ], [Dη]) associated to consecutive rays in the normal fan N from the left-most picture
in Figure 5.3. Such numbers are computed as the quotient of the gcd of maximal minors of the matrix ([Dρ] | [Dη]) by the determinant of the 2 × 2-matrix (ρ | η). In our example,
all such numbers equal one. In particular, this shows why we need not consider the fanN 0 and, instead, work safely with the fan N .
P1⊕ P2 P2⊕ P3 P1⊕ P3
Figure 5.4: From left to right: Minkowski sum of the Newton polytopes P1+ P2, P2 + P3 and
P1+ P3.
The resulting weighted graph has four bivalent nodes (in gray) and it is depicted on the right of Figure 5.5. After removing these gray nodes, we obtain a graph with f -vector (7, 13). The complement of the graph has eight connected components. Notice that the nodes e2, [D1] = [D4], [D3] and [D5] are aligned in the picture. This reflect the fact that
these four vectors generate a two-dimensional cone in R3. In addition to the four bivalent
nodes, this also explains the difference between the number of edges in the tropical surface graph and the number of edges in the abstract graph, seen on the left-side of Figure 5.5. In particular, the predicted edge ([D4], [D5]) can be seen in the graph as the line segment
containing the points [D4], [D3] and [D5].
We certify our calculations by computing the Newton polytope of the generator of the principal ideal (x − f1(s, t), y − f2(s, t), z − f3(s, t)) ∩ C[x, y, z] using Singular [21]. For
generic choices of coefficients a1, . . . , c3, this polynomial has degree 14. Its Newton polytope
Figure 5.5: From left to right: weighted simplicial complexes representing T Y . The left one corresponds to the abstract graph and the right one is the planar graph obtained by realizing the abstract graph and combining weights of overlapping edges. The broken edges on the left graph have weight zero and they dissapear in the planar graph.
RAYS -1 0 0 # 0 0 -1 0 # 1 1 1 1 # 2 0 0 -1 # 3 2 1 2 # 4 3 2 3 # 5 5 3 4 # 6 N_RAYS 7 F_VECTOR 1 7 13 MAXIMAL_CONES {0 1} # Dimension 3 {0 2} {0 3} {0 4} {0 5} {5 6} {2 3} {4 5} {1 4} {2 5} {1 3} {4 6} {3 6} MULTIPLICITIES 4 # Dimension 3 1 6 1 1 1 1 2 4 1 4 1 1
After changing the signs of the seven rays in the left-most column to overcome the max convention of Gfan, we recover the nodes e1, e2, [D5], e3, [D1], [D3] and [D2], as expected.
Example 5.3.4. We consider the morphism f = (f1, f2, f3) : C2 → Y ⊂ C3 given by
f1(s, t) = a1s2+ a2s3+ a3t2, f2(s, t) = b1t2+ b2t3+ b3s2, f3(s, t) = c1st + c2s3+ c3t3+ c4st2+ c5s2t,
with generic coefficients a1, . . . , c5 ∈ C∗. By construction, the degree of the map f equals
one. As in the previous example, we draw the Newton polytopes of each fi (Figure 5.6),
(Figure 5.8). In this case, ray r7 is a common ray of two of the constituent fans. We indicate
this phenomenon on the right-most picture in Figure 5.8 by drawing a purple edge, which combines the red and blue edges from the polytopes P1 and P2. Similarly, the purple ray
in the normal fan on the left-most picture in Figure 5.8 combines the red and blue rays of N (P1) and N (P2). In addition, three rays out of the nine rays in W3i=1N (Pi) have
non-trivial weights as indicated in the pictures on Figure 5.7.
P1 v3 v1 v2 (0, 2) (2, 0) (3, 0) P2 v3 v1 v2 (0, 2) (0, 3) (2, 0) P3 v1 v2 v3 (1, 1) (3, 0) (0, 3)
Figure 5.6: From left to right: Newton polytopes of the polynomials f1, f2 and f3.
N (P1) v1 v2 v3 (1, 1) (0, 1) (−2, −3) 2 N (P2) v1 v2 v3 (1, 1) (0, 1) (−2, −3) 2 N (P3) v1 v2 v3 (−1, −1) (2, 1) (1, 2) 3
Figure 5.7: From left to right: inner normal fans of the Newton polytopes P1, P2 and P3.
Following the notation of Figure 5.8, the nodes of the tropical surface graph have co- ordinates: e1, e2, e3, [D1] = (0, 0, 0), [D2] = (−9, −6, −9), [D3] = (−3, −3, −3), [D4] =
(−6, −9, −9), [D5] = (0, 0, 0), [D6] = (2, 2, 3), [D6] = (2, 2, 3), [D7] = (2, 2, 2) and [D8] =
(2, 2, 3). After going through dimension testings, we obtain a list of fourteen edges: three with |J | = 2, (ei, ej), seven with |J | = 1, ([D4], e1), ([D7], e1), ([D2], e2), ([D7], e2), ([D3], e3),
([D6], e3), ([D8], e8) and four with |J | = 0, ([D2], [D3]), ([D3], [D4]), ([D6], [D7]), ([D7], [D8]).
Finally, following the approach of the previous example, we compute tropical multiplicities as mixed volumes. The transition from the weighted abstract graph to its realization is